Bayesian sensor estimation for machine condition monitoring

ABSTRACT

A method for monitoring a system includes receiving a set of training data. A Gaussian mixture model is defined to model a probability distribution for a particular sensor of the system from among a plurality of sensors of the system based on the received training data. The Gaussian mixture model includes a sum of k mixture components, where k is a positive integer. Sensor data is received from the plurality of sensors of the system. An expectation-maximization technique is performed to estimate an expected value for the particular sensor based on the defined Gaussian mixture model and the received sensor data from the plurality of sensors.

CROSS-REFERENCE TO RELATED APPLICATION

The present application is based on provisional application Ser. No.60/849,700, filed Oct. 5, 2006, the entire contents of which are hereinincorporated by reference.

BACKGROUND OF THE INVENTION

1. Technical Field

The present disclosure relates to machine condition monitoring and, morespecifically, to Bayesian sensor estimation for machine conditionmonitoring.

2. Discussion of the Related Art

Condition monitoring relates to the observation and analysis of one ormore sensors that sense key parameters of machinery. By closelyobserving the sensor data, a potential failure or inefficiency may bedetected and remedial action may be taken, often before a major systemfailure occurs.

Effective condition monitoring may allow for increased uptime, reducedcosts associated with failures, and a decreased need for prophylacticreplacement of machine components.

Condition monitoring may be applied to a wide variety of industrialmachinery such as capitol equipment, factories and power plants;however, condition monitoring may also be applied to other mechanicalequipment such as automobiles and non-mechanical equipment such ascomputers. In fact, principals of condition monitoring may be appliedmore generally to any system or organization. For example, principals ofcondition monitoring may be used to monitor the vital signs of a patientto detect potential health problems. For example, principals ofcondition monitoring may be applied to monitor performance and/oreconomic indicators to detect potential problems with a corporation oran economy.

In condition monitoring, one or more sensors may be used. Examples ofcommonly used sensors include vibration sensors for analyzing a level ofvibration and/or the frequency spectrum of vibration. Other examples ofsensors include temperature sensors, pressure sensors, spectrographicoil analysis, ultrasound, and image recognition devices.

A sensor may be a physical sensory device that may be mounted on or neara monitored machine component or a sensor may more generally refer to asource of data.

Conventional techniques for condition monitoring acquire data from theone or more sensors and analyze the collected data to detect when thedata is indicative of a potential fault. Inferential sensing is anexample of an approach that may be used to determine when sensor data isindicative of a potential fault.

In inferential sensing, an expected value for a particular sensor{circumflex over (x)} is estimated, for example, through the use ofother observed sensors y. The actual sensor value y may then be comparedto the expected sensor value {circumflex over (x)}, and the larger thedifference between the two values, the greater the likelihood of apotential fault.

Accordingly, fault diagnosis is typically performed in two steps. In thefirst step, based on observed sensor values y, the expected particularsensor value {circumflex over (x)} is calculated. This step is known as“sensor estimation.” The “residue” is defined as the difference betweenthe estimated value {circumflex over (x)} and the observed value y.Then, in a “rule-based decision step,” the values of y, {circumflex over(x)} and the residue are analyzed with respect to a set of rules toproperly identify the presence of and type of a potential fault.

The set of rules are generally defined by experts familiar with thecorrect operations of the system being monitored.

In the sensor estimation step, a variety of techniques may be used toprovide the estimated value {circumflex over (x)} based on the set ofobserved sensor values y. Such approach may involve the use ofmultivariate state estimation techniques (MSET) or auto-associate neuralnetworks (AANN). Details concerning these approaches may be found, forexample, in J. W. Hines, A. Gribok and B. Rasmussen (2001), “On-LineSensor Calibration Verification: A survey,” International Congress andExhibition on Condition Monitoring and Diagnostic EngineeringManagement, incorporated herein by reference.

In each of these methods, involve the building ofmultiple-input-multiple-output networks to calculate {circumflex over(x)} based on the set of observed sensor values y. The networksthemselves may be established base on a set of training data thatincludes the sensor values X and Y observed during fault-free operationof the system being monitored. Later, when the system being monitored isbrought on-line, observed faults may be difficult to detect because thecharacteristics of the observed faults would not have been observedduring the period of collection of the training data.

Accordingly, artificial faults may be added to the training data as isdone in D. Wrest, J. W. Hines, and R. E. Uhrig (1996), “InstrumentSurveillance and Calibration Verification Through Plant Wide MonitoringUsing Autoassociative Neural Networks,” The American Nuclear SocietyInternational Topical Meeting on Nuclear Plant Instrumentation, Controland Human Machine Interface Technologies, May 6-9, 1996, incorporatedherein by reference. However, it may be difficult or impossible toobtain artificial fault data for every conceivable fault, given thenumber of possible deviations and the high dimensionality of the sensorvector.

Another approach to providing the estimated value {circumflex over (x)}based on the set of observed sensor values y involves support vectorregression (SVR). Examples of such approaches may be found in A. V.Gribok, J. W. Hines and R. E. Uhrig (2000), “Use of Kernel BasedTechniques for Sensor Validation in Nuclear Power Plants,” InternationalTopical Meeting on Nuclear Plant Instrumentation, Controls andHuman-Machine Interface Technologies, incorporated herein by reference.

In SVR, an estimate for a particulate sensor X_(i) ({circumflex over(x)}_(i)) is determined from the observed values of other sensorsy_(j≠i). SVR makes the basic assumption that X_(i) is predictable fromy_(j≠i), however, this assumption often does not hold true. For example,in a power plant, a process driver inlet temperature is not accuratelypredictable based on any other sensor values. Additionally, a faultwithin the sensors y_(j≠i) may result in an inaccurate estimation forX_(i), even if the i-th sensor is normal.

SUMMARY

A method for monitoring a system includes receiving a set of trainingdata. A Gaussian mixture model is defined to model a probabilitydistribution for a particular sensor of the system from among aplurality of sensors of the system based on the received training data.The Gaussian mixture model includes a sum of k mixture components, wherek is a positive integer. Sensor data is received from the plurality ofsensors of the system. An expectation-maximization technique isperformed to estimate an expected value for the particular sensor basedon the defined Gaussian mixture model and the received sensor data fromthe plurality of sensors.

An actual sensor value may be received from the particular sensor. Thereceived actual sensor value may be compared to the estimated expectedsensor value. A potential fault may be detected based on the comparison.

Performing the expectation-maximization technique may include rankingthe k mixture components according to a degree of influence on theGaussian mixture model, selecting a set of mixture components that aremost influential, iteratively improving an estimation of the expectedvalue for the particular sensor based on a diagonal covariance matrixthat contributes to a relationship between the received sensor data fromthe plurality of sensors and the expected value for the particularsensor, using the selected set of mixture components, iterativelyimproving an estimation of the diagonal covariance matrix using theselected set of mixture components, and repeating the steps ofiteratively improving the estimation of the expected value for theparticular sensor and iteratively improving the estimation of thediagonal covariance matrix until the two estimates achieve convergence.

Convergence may be achieved by repeating the steps of iterativelyimproving the estimation of the expected value for the particular sensorand iteratively improving the estimation of the diagonal covariancematrix until subsequent improvements are negligible.

Performing the expectation-maximization technique may include estimatingthe expected value for the particular sensor, and estimating a diagonalcovariance matrix that contributes to a relationship between thereceived sensor data from the plurality of sensors and the expectedvalue for the particular sensor, wherein the two estimations areperformed at substantially the same time.

The training data may include sensor data from the plurality of sensorsand sensor data from the particular sensor taken during a period offault-free operation of the system.

Each of the k mixture components may be a Gaussian distribution definedby a mean and a variance that are determined during the performance ofthe expectation-maximization technique.

A system for monitoring a machine includes a plurality of sensors formonitoring the machine including a particular sensor. A Gaussian mixturemodel defining unit defines a Gaussian mixture model to for estimatingan expected value for the particular sensor. The Gaussian mixture modelincludes a sum of a plurality of mixture components. An estimation unitestimates the expected value for the particular sensor based on thedefined Gaussian mixture model.

Training data may be used by the Gaussian mixture model defining unit indefining the plurality of mixture components.

The training data may include sensor data from the plurality of sensorsand sensor data from the particular sensor taken during a period offault-free operation of the machine.

An expectation-maximization technique may be used to estimate anexpected value for the particular sensor based on the defined Gaussianmixture model and the received sensor data from the plurality ofsensors.

Performing the expectation-maximization technique may include rankingthe plurality of mixture components according to a degree of influenceon the Gaussian mixture model, selecting a set of mixture componentsthat are most influential, iteratively improving an estimation of thediagonal covariance matrix using the selected set of mixture components,and repeating the steps of iteratively improving the estimation of theexpected value for the particular sensor and iteratively improving theestimation of the diagonal covariance matrix until the two estimatesachieve convergence.

Convergence may be achieved when repeating the steps of iterativelyimproving the estimation of the expected value for the particular sensorand iteratively improving the estimation of the diagonal covariancematrix results in a negligible improvement.

Performing the expectation-maximization technique may include estimatingthe expected value for the particular sensor, and estimating a diagonalcovariance matrix that contributes to a relationship between thereceived sensor data from the plurality of sensors and the expectedvalue for the particular sensor. The two estimations may be performed atsubstantially the same time.

A computer system includes a processor and a program storage devicereadable by the computer system, embodying a program of instructionsexecutable by the processor to perform method steps for monitoring asystem. The method includes receiving a set of training data. A Gaussianmixture model is defined to model a probability distribution for aparticular sensor of the system from among a plurality of sensors of thesystem based on the received training data. The Gaussian mixture modelincludes a sum of k mixture components, where k is a positive integer.Sensor data is received from the plurality of sensors of the system, andan expectation-maximization technique is performed to estimate anexpected value for the particular sensor based on the defined Gaussianmixture model and the received sensor data from the plurality ofsensors. Each of the k mixture components is a Gaussian distributiondefined by a mean and a variance that are determined during theperformance of the expectation-maximization technique.

An actual sensor value may be received from the particular sensor. Thereceived actual sensor value may be compared to the estimated expectedsensor value. A potential fault may be detected based on the comparison.

Performing the expectation-maximization technique may include rankingthe k mixture components according to a degree of influence on theGaussian mixture model. A set of mixture components that are mostinfluential may be selected. An estimation of the expected value for theparticular sensor may be iteratively improved based on a diagonalcovariance matrix that contributes to a relationship between thereceived sensor data from the plurality of sensors and the expectedvalue for the particular sensor, using the selected set of mixturecomponents. An estimation of the diagonal covariance matrix may beiteratively improved using the selected set of mixture components. Thesteps of iteratively improving the estimation of the expected value forthe particular sensor and iteratively improving the estimation of thediagonal covariance matrix may be repeated until the two estimatesachieve convergence.

Convergence may be achieved when repeating the steps of iterativelyimproving the estimation of the expected value for the particular sensorand iteratively improving the estimation of the diagonal covariancematrix results in a negligible improvement.

Performing the expectation-maximization technique may include estimatingthe expected value for the particular sensor, and estimating a diagonalcovariance matrix that contributes to a relationship between thereceived sensor data from the plurality of sensors and the expectedvalue for the particular sensor. The two estimations may be performed atsubstantially the same time.

The training data may include sensor data from the plurality of sensorsand sensor data from the particular sensor taken during a period offault-free operation of the system.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the present disclosure and many of theattendant advantages thereof will be readily obtained as the samebecomes better understood by reference to the following detaileddescription when considered in connection with the accompanyingdrawings, wherein:

FIG. 1 is a flow chart illustrating an approach to system conditionmonitoring according to an exemplary embodiment of the presentinvention;

FIG. 2 is a graph showing sensor data for sensor BPTC6B from the400^(th) data point to the 800^(th) data point;

FIG. 3 illustrates results of the GMM approach according to an exemplaryembodiment of the present invention as compared to the SVR and MSETapproaches during a sensor fault condition;

FIG. 4 illustrates the results of the GMM approach according to anexemplary embodiment of the present invention as compared to the SVR andMSET approaches without a sensor fault condition; and

FIG. 5 shows an example of a computer system capable of implementing themethod and apparatus according to embodiments of the present disclosure.

DETAILED DESCRIPTION OF THE DRAWINGS

In describing the exemplary embodiments of the present disclosureillustrated in the drawings, specific terminology is employed for sakeof clarity. However, the present disclosure is not intended to belimited to the specific terminology so selected, and it is to beunderstood that each specific element includes all technical equivalentswhich operate in a similar manner.

Exemplary embodiments of the present invention utilize a Bayesianframework to determine a general-case probability distribution fordetermining an expected value for X from detected sensor data Y. AGaussian mixture model (GMM) may be used to model the probabilitydistribution of the potential values for X. A GMM is an approach tocharacterizing a distribution, here the potential values of X, as acombination of other constituent distributions, each of which may be aGaussian distribution.

These constituent distributions may be considered mixture components,and each mixture component may be a Gaussian distribution. The completeprobability distribution for x may be a combination of k such mixturecomponents, where k is a positive integer.

Here, a random Gaussian vector ε is introduced to represent the possiblerelationship between Y and X. Thus, ε incorporates a covariance matrixthat can transform Y into an expected value for X. The covariance matrixof ε is assumed to be an unknown parameter and may be estimatedadaptively for each input Y.

In using a Gaussian mixture model (GMM) to characterize the probabilitydistribution of x, the following expression may be used to:

$\begin{matrix}{{P(x)} = {\sum\limits_{s = 1}^{k}{{P\left( x \middle| s \right)}{P(s)}}}} & (1)\end{matrix}$wherein P(x) represents the probability distribution of X and P(x) isdefined as a sum of k mixture components from s=1 to s=k. Thus s is thelabel for the s-th mixture component, P(s) is the probabilitydistribution based on the s-th mixture component, and P(x|s) is theprobability distribution of X given the s-th mixture component. Theprobability distribution for each P(x|s) may have a Gaussiandistribution and may be expressed P(x|s)˜N(m_(x|s), c_(x|s)), wherem_(x|s) is the mean and c_(x|s) is the covariance of the Gaussiandistribution for the s-th mixture component.

Accordingly, the relationship between y, x, and ε may be expresses interms of the deviation vector ε as follows:y=x+ε  (2)

Initially, ε may be set to have a Gaussian distribution with a mean ofzero and an unknown diagonal covariance matrix Θ. Accordingly, ε˜N(0,Θ).The dimensionality of x may be expressed as d and thus Θ has d freeparameters.

Of the complete set of i sensors, the diagonal covariance matrix Θincludes the variance of deviation for each sensor i, with each varianceof deviation expressed as Θ_(i). When an i-th sensor is normal, thedeviation Θ_(i) is small indicating that x_(i) is close to y_(i).Meanwhile, when the i-th sensor is faulty, Θ_(i) is large indicatingthat x_(i) is not close to y_(i). Because the diagonal covariance matrixΘ comprises a set of variances that each depend on the observed sensordata, Θ is adaptive rather than fixed.

As described above, the actual sensor value of a particular monitoredsensor is x, y is the full set of observed sensor data in the monitoredsystem, and {circumflex over (x)} is the expected value for x.Accordingly, the expected value {circumflex over (x)} is the conditionalexpectation of x given y:{circumflex over (x)}=E(x|y)  (3)

Given equation (1), the conditional expectation may be written as a GMMas follows:

$\begin{matrix}{{E\left( x \middle| y \right)} = {\sum\limits_{s = 1}^{k}{{P\left( s \middle| y \right)}{E\left( {\left. x \middle| y \right.,s} \right)}}}} & (4)\end{matrix}$

However, because Θ is adaptive and initially unknown, both Θ and{circumflex over (x)} are estimated and thus the GMM is determined withtwo unknowns. Exemplary embodiments of the present invention estimateboth Θ and {circumflex over (x)} substantially simultaneously.

An expectation-maximization (EM) approach may be used to estimate{circumflex over (x)} and Θ at substantially the same time. A moredetailed explanation of EM approaches may be found in A. P. Dempster, N.M. Laird and D. B. Rubin (1977), “Maximum-likelihood from IncompleteData via the EM Algorithm,” Journal of the Royal Statistical Society,Series B, 39, pp. 1-38, which is hereby incorporated by reference.

EM approaches allow for maximum likelihood estimation of a parameterwhen variables are hidden. Here, {circumflex over (x)} may be viewed asthe hidden variable and Θ may be the parameter to be estimated. As usedherein, the EM approach includes an expectation step and a maximizationstep. In the expectation step, {circumflex over (x)} may be estimated.In the maximization step, Θ may be estimated. These steps arealternately performed until convergence is achieved. Thus eachestimation is refined and the refinement of one estimation is used torefine the other estimation. In this way, it is said that bothestimations are refined at substantially the same time.

For example, in the expectation step, the current estimate of Θ is usedto compute the estimated value for {circumflex over (x)}, for example,using equation (3) above.

Then, in the maximization step, the variance of deviation for the s-thGaussian mixture component, Θ_(s) may be estimated and this value iscombines as follows to result in an improved estimate for Θ:

$\begin{matrix}{\Theta = {\sum\limits_{s = 1}^{K}{{P\left( s \middle| y \right)}\Theta_{s}}}} & (5)\end{matrix}$

The expectation step and the maximization step may then be alternatelyrepeated until convergence is achieved. Convergence is achieved wheneach additional step results in only a negligible change.

Each iterative step of the EN approach may involve complex calculations,including calculating the inverse of the covariance matrix c_(x|s) foreach Gaussian component, thus when the number of sensors d is large,real-time monitoring may be difficult as the complexity of the EMcalculations are computationally expensive.

Exemplary embodiments of the present invention utilize an isotropicGaussian Model to model each mixture component. Under this approach,P(x|s)˜N(m_(x|s), σ²I_(d)), where the covariance matrix is themultiplication of a scalar σ² and a d×d identity matrix I_(d).Accordingly, the complexity of the EM calculation may be substantiallyreduced and speed may be increased to more easily perform real-timemonitoring.

In application, not all mixture components make a substantialcontribution to the estimation {circumflex over (x)} and Θ. For example,all but several components may have a negligible influence on theestimations. Meanwhile, it may be computationally expensive to process alarge number of mixture components. Accordingly, during the firstiteration of the EM approach, the probability distribution P(s|y) foreach mixture component may be ranked according to their degree ofinfluence within the complete mixture model. For example, the mixturecomponents may be ranked from highest impact to lowest impact. Then,starting with the highest ranked component, components are selecteduntil the sum of the selected components exceeds a desired level. Forexample, mixture components may be selected until the cumulativeinfluence of the selected mixture components equals or exceeds a 95%influence. Thereafter, only the selected components are used insubsequent EM iterations. In practice, the number of mixture componentsmay be significantly reduced and accordingly, the speed of EMcalculations may be further increased.

As discussed above, the Gaussian mixture model for estimating x may becalculated based on training data. The Gaussian mixture model mayinclude a set of mixture components k, and may be a fixed value, forexample 60. Each mixture component, as a Gaussian function, may bedefined in terms of a component center (mean) m_(x|s) and a variance σ².In calculating each optimal mixture component, these attributes may bedetermined. For example, the component center may be determined using ak-mean algorithm and the variance σ² may be determined using aleave-one-out process that maximizes the log likelihood of P(x), forexample, by computing multiple possible solutions and the using thesolution that proves to be most useful.

The leave-one-out process may produce a probability distribution P(x)with a greater generalization performance than may otherwise be obtainedby applying maximum likelihood estimation to the entire training set.

After training is complete, the input y is collected. Then, theexpectation-maximization (EM) approach may be used to estimate{circumflex over (x)} and Θ at substantially the same time. Such anapproach involves the iterative improvement of both unknowns; howeverinitial values may first be provided. In determining an initial valuefor {circumflex over (x)}, {circumflex over (x)} may be initialized byreceiving a value equal to the closest component center to the input y.Then, during the initial EM iteration step, equations (4) and (5) may becomputed based on all of the k mixture components.

Next, mixture components may be ranked and selected as described above.Subsequent EM iteration steps are then performed with only the selectedmixture components. Then, the estimate {circumflex over (x)} may bereturned and compared with y. Finally, a potential fault may be detectedby comparing {circumflex over (x)}, y, and the residue (y−{circumflexover (x)}) in the rule-based decision step.

FIG. 1 is a flow chart illustrating an approach to system conditionmonitoring according to an exemplary embodiment of the presentinvention. First, training data 31 may be obtained (Step S510). Thetraining data may be obtained, for example, by running the system undermonitoring 33 during normal conditions and recording sensor data.Alternatively, the training data 31 may be provided from a training datadatabase (not shown) or over a computer network (not shown). Thetraining data may have been produced either the system under monitoring33 or a similar system. Once the system under monitoring is broughton-line and is functioning normally, data obtained during the system'snormal operation may be added to the set of training data. Thus, as themonitoring continues under normal conditions, the set of training datamay be increased and accordingly, accuracy may increase with use.

Next, P(x), the probability distribution for x, may be defined as aGaussian mixture model (GEM) comprising a sum of k mixture components(Step S11), for example, P(x) may be calculated using equation (1)discussed above. The k mixture components may then each be determinedaccording to a component center (mean) of m_(x|s) and a variance σ²(Step S12). Then, the condition of the system may be monitored while thesystem is on-line and operational. Monitoring of the system may beperformed by one or more sensors. Sensor data 32 (collectively referredto as y) may be collected (Step S13).

An expectation-maximization (EM) process may then be performed togeneralize the relationship between an expected value of a particularsensor {circumflex over (x)} and the sensor data y (Step S14). The EMprocess may include multiple steps, for example, first, the k mixturecomponents may be ranked, as described above (Step S20) and then themost influential mixture components are selected (Step S21).Accordingly, in subsequent EM steps, processing may be expedited bylimiting computation to the selected mixture components. Then,estimation of the parameter Θ and the variable {circumflex over (x)} mayoccur substantially at the same time by alternatively performing anexpectation step (Step S22) where the estimation of {circumflex over(x)} is calculated based on the latest calculated value for Θ, and amaximization step (Step S23) where the variance of deviation for thes-th Gaussian mixture component Θ_(s) is added to Θ.

The expectation (Step S22) and maximization (Step S23) steps may then berepeated for as long as {circumflex over (x)} and Θ have not achievedconvergence (No, Step S24). Convergence is achieved when subsequentiterations of both {circumflex over (x)} and Θ no longer providesubstantial changes.

After the EM process has achieved convergence (Yes, Step S24), the EMprocess has completed (Step S14) and {circumflex over (x)} may beestimated (Step S15) based on y and the generalized relationship between{circumflex over (x)} and y that was determined during the EM process.Finally, the resulting values of {circumflex over (x)} and y and thedifference between them (the variance) may be considered as part of arule-based decision step where potential faults are detected (Step 516).

In this step, a potential fault is detected (Yes, Step S16) when thepredetermined rules are satisfied based on the values of {circumflexover (x)}, y and the variance. When a potential fault is detected (Yes,Step S16), an alert may be generated (Step S17). When a potential faultis not detected (No, Step S17), monitoring of the system may continuewith the collection of the sensor data y (Step S13).

Exemplary embodiments of the present invention may be testedexperimentally. In one such experiment, 35 sensors are placed to monitora system. The sensors include a gas flow sensor, an inlet temperaturesensor, an IGV actuator position sensor, and 32 blade path temperaturesensors named BPTC1A, BPTC1B, BPTC2A, . . . , BPTC16B. The system isbrought on-line and 2000 data points are collected. Of these, the first360 data points are used as training data and the remaining 1640 datapoints are used in testing.

In this example, the blade path temperature sensor BPTC6B becomes faultybetween data points 500 and 600. This fault is represented as anincrease of 30° between data points 500 and 600. FIG. 2 is a graphshowing sensor data for sensor BPTC6B from the 400^(th) data point tothe 800^(th) data point. The numbered data points are shown along theabscissa while the sensed temperature in degrees is shown along theordinate.

Based on the sensor data, a Gaussian mixture model (GMM) approach inaccordance with an exemplary embodiment of the present invention may beutilized to estimate a value for each sensor. For comparison, techniquesfor estimating sensor values based on support vector regression (SVR)and multivariate state estimation techniques (MSET) may also be used.Based on these approaches, expected sensor values may be calculated andactual sensor values collected, the residues may then be calculatedbased on the difference between expected and actual sensor values.

During normal operation of the system, residues should be relativelysmall and during a fault condition, residues should be relatively large.The greater this distinction is, the easier it is to detect a fault.Because the test data includes both normal operation and a faultcondition, the test data may be used to gauge the effectiveness of theexemplary embodiment of the present invention (the GMM approach) againstthe SVR and MSET approaches being tested.

FIG. 3 illustrates results of the GMM approach according to an exemplaryembodiment of the present invention as compared to the SVR and MSETapproaches during a sensor fault condition. FIG. 3 shows the residue,the difference between the expected sensor value and actual sensor valuefor the sensor BPTC6B that is faulty between data point 500 and 600. Ascan be seen from this graph, the GMM approach shows a relatively smalland consistent residue while the sensor is not faulty from data points400 to 500 and from 600 to 800. However, from data points 500 to 600,while the sensor is faulty, the GMM approach shows a relatively largeand consistent residue.

FIG. 4 illustrates the results of the GMM approach according to anexemplary embodiment of the present invention as compared to the SVR andMSET approaches without a sensor fault condition. FIG. 4 shows theresidue for sensor BPTC1B from data points 400 to 800. There are nosensor faults for this sensor. As can be seen from this graph, the GMMapproach shows a relatively small and consistent residue throughout theentire reading. As large residues during fault-free operation are morelikely to lead to a false alarm, the relatively small and consistentresidue of the GMM approach represents an increased level of precisionover the other approaches shown.

FIG. 5 shows an example of a computer system which may implement amethod and system of the present disclosure. The system and method ofthe present disclosure may be implemented in the form of a softwareapplication running on a computer system, for example, a mainframe,personal computer (PC), handheld computer, server, etc. The softwareapplication may be stored on a recording media locally accessible by thecomputer system and accessible via a hard wired or wireless connectionto a network, for example, a local area network, or the Internet.

The computer system referred to generally as system 1000 may include,for example, a central processing unit (CPU) 1001, random access memory(RAM) 1004, a printer interface 1010, a display unit 1011, a local areanetwork (LAN) data transmission controller 1005, a LAN interface 1006, anetwork controller 1003, an internal bus 1002, and one or more inputdevices 1009, for example, a keyboard, mouse etc. As shown, the system1000 may be connected to a data storage device, for example, a harddisk, 1008 via a link 1007.

The above specific exemplary embodiments are illustrative, and manyvariations can be introduced on these embodiments without departing fromthe spirit of the disclosure or from the scope of the appended claims.For example, elements and/or features of different exemplary embodimentsmay be combined with each other and/or substituted for each other withinthe scope of this disclosure and appended claims.

1. A method for monitoring a system, comprising: receiving a set oftraining data; using a processor for defining a Gaussian mixture modelto model a probability distribution for a particular sensor of thesystem from among a plurality of sensors of the system based on thereceived training data, the Gaussian mixture model comprising a sum of kmixture components, wherein k is a positive integer; receiving sensordata from the plurality of sensors of the system; and performing anexpectation-maximization technique to estimate an expected value for theparticular sensor based on the defined Gaussian mixture model and thereceived sensor data from the plurality of sensors, wherein each of thek mixture components is a Gaussian distribution defined by a mean and avariance that are determined during the performance of theexpectation-maximization technique.
 2. The method of claim 1,additionally comprising: receiving an actual senor value from theparticular sensor; comparing the received actual sensor value to theestimated expected sensor value; and detecting a potential fault basedon the comparison.
 3. The method of claim 1, wherein the training datacomprises sensor data from the plurality of sensors and sensor data fromthe particular sensor taken during a period of fault-free operation ofthe system.
 4. A method for monitoring a system, comprising: receiving aset of training data; using a processor for defining a Gaussian mixturemodel to model a probability distribution for a particular sensor of thesystem from among a plurality of sensors of the system based on thereceived training data, the Gaussian mixture model comprising a sum of kmixture components, wherein k is a positive integer; receiving sensordata from the plurality of sensors of the system: and performing anexpectation-maximization technique to estimate an expected value for theparticular sensor based on the defined Gaussian mixture model and thereceived sensor data from the plurality of sensors, wherein performingthe expectation-maximization technique comprises: ranking the k mixturecomponents according to a degree of influence on the Gaussian mixturemodel; selecting a set of mixture components that are most influential;iteratively improving an estimation of the expected value for theparticular sensor based on a diagonal covariance matrix that contributesto a relationship between the received sensor data from the plurality ofsensors and the expected value for the particular sensor, using theselected set of mixture components; iteratively improving an estimationof the diagonal covariance matrix using the selected set of mixturecomponents; and repeating the steps of iteratively improving theestimation of the expected value for the particular sensor anditeratively improving the estimation of the diagonal covariance matrixuntil the two estimates achieve convergence.
 5. The method of claim 4,wherein convergence is achieved when repeating the step of iterativelyimproving the estimation of the expected value for the particular sensorand iteratively improving the estimation of the diagonal covariancematrix results in a negligible improvement.
 6. A method for monitoring asystem, comprising: receiving a set of training data; using a processorfor defining a Gaussian mixture model to model a probabilitydistribution for a particular sensor of the system from among aplurality of sensors of the system based on the received training data,the Gaussian mixture model comprising a sum of k mixture components,wherein k is a positive integer; receiving sensor data from theplurality of sensors of the system; and performing anexpectation-maximization technique to estimate an expected value for theparticular sensor based on the defined Gaussian mixture model and thereceived sensor data from the plurality of sensors, wherein performingthe expectation-maximization technique comprises: estimating theexpected value for the particular sensor; and estimating a diagonalcovariance matrix that contributes to a relationship between thereceived sensor data from the plurality of sensors and the expectedvalue for the particular sensor wherein the two estimations areperformed at substantially the same time.
 7. A system for monitoring amachine, comprising: a plurality of sensors for monitoring the machineincluding a particular sensor; a Gaussian mixture model defining unitfor defining a Gaussian mixture model to for estimating an expectedvalue for the particular sensor, the Gaussian mixture model comprising asum of a plurality of k mixture components, wherein k is a positiveinteger; and an estimation unit for estimating the expected value forthe particular sensor based on the defined Gaussian mixture model,wherein each of the k mixture components is a Gaussian distributiondefined by a mean and a variance that are determined during theperformance of estimating the expected value for the particular sensor.8. The system of claim 7, additionally comprising training data for useby the Gaussian mixture model defining unit in defining the plurality ofmixture components.
 9. The system of claim 8, wherein the training datacomprises sensor data from the plurality of sensors and sensor data fromthe particular sensor taken during a period of fault-free operation ofthe machine.
 10. The system of claim 7, wherein anexpectation-maximization technique is used to estimate an expected valuefor the particular sensor based on the defined Gaussian mixture modeland the received sensor data from the plurality of sensors.
 11. Thesystem of claim 10, wherein performing the expectation-maximizationtechnique comprises: estimating the expected value for the particularsensor; and estimating a diagonal covariance matrix that contributes toa relationship between the received sensor data from the plurality ofsensors and the expected value for the particular sensor, wherein thetwo estimations are performed at substantially the same time.
 12. Asystem for monitoring a machine, comprising: a plurality of sensors formonitoring the machine including a particular sensor; a Gaussian mixturemodel defining unit for defining a Gaussian mixture model to forestimating an expected value for the particular sensor, the Gaussianmixture model comprising a sum of a plurality of mixture components; andan estimation unit for estimating the expected value for the particularsensor based on the defined Gaussian mixture model, wherein anexpectation-maximization technique is used to estimate an expected valuefor the particular sensor based on the defined Gaussian mixture modeland the received sensor data from the plurality of sensors, and whereinperforming the expectation-maximization technique comprises: ranking theplurality of mixture components according to a degree of influence onthe Gaussian mixture model; selecting a set of mixture components thatare most influential; iteratively improving an estimation of thediagonal covariance matrix using the selected set of mixture components;and repeating the steps of iteratively improving the estimation of theexpected value for the particular sensor and iteratively improving theestimation of the diagonal covariance matrix until the two estimatesachieve convergence.
 13. The system of claim 12, wherein convergence isachieved when repeating the steps of iteratively improving theestimation of the expected value for the particular sensor anditeratively improving the estimation of the diagonal covariance matrixresults in a negligible improvement.
 14. A computer system comprising: aprocessor; and a program storage device readable by the computer system,embodying a program of instruction executable by the processor toperform method steps for monitoring a system, the method comprising:receiving a set of training data; defining a Gaussian mixture model tomodel a probability distribution for a particular sensor of the systemfrom among a plurality of sensors of the system based on the receivedtraining data the Gaussian mixture model comprising a sum of k mixturecomponents, wherein k is a positive integer; receiving sensor data fromthe plurality of sensors of the system; and performing anexpectation-maximization technique to estimate an expected value for theparticular sensor based on the defined Gaussian mixture model and thereceived sensor data from the plurality of sensors, wherein each of thek mixture components is a Gaussian distribution defined by a mean and avariance that are determined during the performance of theexpectation-maximization technique.
 15. The computer system of claim 14,additionally comprising: receiving an actual senor value from theparticular sensor; comparing the received actual sensor value to theestimated expected sensor value; and detecting a potential fault basedon the comparison.
 16. The computer system of claim 14, whereinperforming the expectation-maximization technique comprises: ranking thek mixture components according to a degree of influence on the Gaussianmixture model; selecting a set of mixture components that are mostinfluential; iteratively improving an estimation of the expected valuefor the particular sensor based on a diagonal covariance matrix thatcontributes to a relationship between the received sensor data from theplurality of sensors and the expected value for the particular sensor,using the selected set of mixture components; iteratively improving anestimation of the diagonal covariance matrix using the selected set ofmixture components; and repeating the steps of iteratively improving theestimation of the expected value for the particular sensor anditeratively improving the estimation of the diagonal covariance matrixuntil the two estimates achieve convergence.
 17. The computer system ofclaim 16 wherein convergence is achieved when repeating the steps ofiteratively improving the estimation of the expected value for theparticular sensor and iteratively improving the estimation of thediagonal covariance matrix results in a negligible improvement.
 18. Thecomputer system of claim 14, wherein performing theexpectation-maximization technique comprises: estimating the expectedvalue for the particular sensor; and estimating a diagonal covariancematrix that contributes to a relationship between the received sensordata from the plurality of sensors and the expected value for theparticular sensor, wherein the two estimations are performed atsubstantially the same time.
 19. The computer system of claim 14,wherein the training data comprises sensor data from the plurality ofsensors and sensor data from the particular sensor taken during a periodof fault-free operation of the system.